Solubility of a family of conics with polynomial coefficients in many variables

Technical Summary: Solubility of a family of conics with polynomial coefficients in many variables

Problem Statement
The paper investigates the asymptotic behavior of the number of rational points on the base of a fibration of conics. Specifically, given three homogeneous polynomials $F_0, F_1, F_2 \in \mathbb{Z}[X_0, \dots, X_n]$ of the same degree $d$, the author studies the family of conics defined by the equation:
$$F_0(\mathbf{y})x_0^2 + F_1(\mathbf{y})x_1^2 = F_2(\mathbf{y})x_2^2$$
where $\mathbf{y} \in \mathbb{P}^n(\mathbb{Q})$ and $x \in \mathbb{P}^2(\mathbb{Q})$. The primary objective is to estimate the quantity $N_{\text{loc}}(\pi_F, B)$, which counts the number of points $\mathbf{y}$ of bounded anticanonical height $H(\mathbf{y}) \leq B$ such that the corresponding conic has a rational point (i.e., is locally soluble everywhere).

This problem is situated within the context of the Loughran–Smeets conjecture (2016) and its refinement by Loughran, Rome, and Sofos (2023). These conjectures predict that for a dominant morphism $\pi: Y \to \mathbb{P}^n$ with geometrically integral generic fiber, the number of locally soluble fibers grows asymptotically as:
$$N_{\text{loc}}(\pi, B) \sim c \frac{B}{(\log B)^{\Delta(\pi)}}$$
where $\Delta(\pi)$ is an invariant determined by the splitting behavior of the fibers over codimension-one points, and $c$ is a constant involving geometric and arithmetic invariants (Tamagawa measures, Brauer groups, etc.).

Methodology
The proof strategy combines several advanced techniques from analytic number theory and arithmetic geometry:

1. Geometric Analysis (Section 2): The author computes the subordinate Brauer group $\text{Br}_{\text{sub}}(\mathbb{P}^n, \pi_F)$ and the invariant $\Delta(\pi_F)$. It is established that for the specific family of conics defined by a "Birch system" of polynomials, the Brauer group quotient is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, and the invariant $\Delta(\pi_F)$ equals $3/2$. This allows for the explicit prediction of the constant $c_{\text{LRS}}(\pi_F)$ predicted by the conjectures.

2. Selberg–Delange Method (Section 3): A general lemma (Lemma 3.1) is proved to estimate the average of arithmetic functions of the form $\tau_z * g$ over polynomials in many variables. This relies on the Taylor expansion of $(s\zeta(s+1))^z$ and provides the necessary tools to handle sums involving multiplicative functions with specific growth conditions.

3. Circle Method and Arithmetic Progressions (Sections 4 & 5): The core analytic work involves estimating the proportion of diagonal plane conics $t_0 x_0^2 + t_1 x_1^2 = t_2 x_2^2$ that possess a rational point, where the coefficients $t_i$ lie in specific arithmetic progressions.

   • The author decomposes the counting function using the Hasse–Minkowski theorem and isolates local solubility conditions.
   • The problem is reduced to estimating sums involving Hilbert symbols and arithmetic functions over coefficients in arithmetic progressions modulo $q$.
   • Theorem 4.2 provides an asymptotic formula for the number of such conics with coefficients in arithmetic progressions, uniformly in the modulus $q$ (provided $q$ is bounded by a power of $\log B$). This result is derived using the Selberg–Delange method applied to the specific arithmetic functions arising from the local solubility conditions.

4. Application of Destagnol–Lyczak–Sofos (Section 6): The author utilizes a recent result by Destagnol, Lyczak, and Sofos (2025) which applies the circle method to estimate the average of arithmetic functions over values of polynomials in many variables. By combining this with the asymptotic formula for conics in arithmetic progressions (Theorem 4.2), the author bridges the gap between the local solubility of the coefficients and the global count of soluble fibers.

Key Contributions and Results

• Theorem 1.7: The main result establishes the Loughran–Rome–Sofos conjecture for the specific family of conics defined by a Birch system of polynomials. Under the assumptions that the varieties cut out by $F_i=0$ are smooth and their intersection is a complete intersection, the author proves:
  $$N_{\text{glob}}(\pi_F, B) \sim c_{\text{LRS}}(\pi_F) \frac{B}{(\log B)^{3/2}}$$
  as $B \to \infty$. Here, $N_{\text{glob}} = N_{\text{loc}}$ due to the Hasse–Minkowski theorem.

• Explicit Constant: The paper provides an explicit formula for the constant $c_{\text{LRS}}(\pi_F)$ (Equation 2.2), confirming it matches the prediction of the conjecture. This constant involves the volume of the region where the conic is soluble over $\mathbb{R}$, a product of local densities over $\mathbb{Q}_p$, and geometric invariants of the base $\mathbb{P}^n$.

• General Lemma (Lemma 3.1): The paper contributes a general estimate for the average of convolutions of multiplicative functions in several variables, which may be applicable to other problems involving the distribution of arithmetic functions over polynomial values.

• Asymptotic for Conics in Progressions (Theorem 4.2): The paper derives a precise asymptotic formula for the number of diagonal conics with coefficients in arithmetic progressions that have a rational point. This result is of independent interest and is crucial for the main proof.

Significance
The paper extends the list of cases where the Loughran–Smeets and Loughran–Rome–Sofos conjectures are known to hold. While previous works had addressed conic bundles over $\mathbb{P}^n$ with specific structures (e.g., Destagnol–Lyczak–Sofos for $D \neq 0$ square-free, Wilson for quadrics over $\mathbb{P}^1 \times \mathbb{P}^1$), this work handles a broader class of families defined by general homogeneous polynomials forming a Birch system.

The significance lies in:

1. Verification of Conjectures: It confirms the predicted order of magnitude $B/(\log B)^{3/2}$ and the explicit shape of the constant for a non-trivial family of conic bundles where the base is $\mathbb{P}^n$ with $n \geq 1$.
2. Methodological Synthesis: It successfully integrates the circle method (via Destagnol–Lyczak–Sofos) with the Selberg–Delange method to handle the arithmetic complexity of local solubility conditions in high-dimensional polynomial families.
3. Refinement of Local-Global Principles: By explicitly calculating the subordinate Brauer group and the splitting invariants, the paper clarifies the geometric obstructions to the existence of rational points in this specific family.

The author notes that the result relies on the "Birch system" condition (Definition 1.6), which ensures the polynomials are sufficiently generic to apply the circle method effectively. The work does not claim to resolve the conjecture for all dominant morphisms but adds a significant class of examples to the verified list.